Method for updating a factor graph of an a posteriori probability estimator

ABSTRACT

A method for updating a factor graph ( 10;10′;10″ ) of an a posteriori probability estimator, the factor graph including at least one repetition node ( 13;13′;13″ ) and at least one sum node ( 11;11′;11″ ), wherein at least two connections are associated with each node, and wherein each connection is associated with an incoming message at the node and with an outgoing message from the node, wherein the method includes the steps of: storing the nodes&#39; incoming and outgoing messages into memory ( 12;12′;12″ ) of the estimator as messages belonging to one same class of wrapped and/or sampled Gaussian messages; updating the node of the factor graph ( 10;10′;10″ ) by using a resulting message belonging to the class of incoming messages, the resulting message being obtained by processing the incoming wrapped and/or sampled Gaussian messages.

BACKGROUND OF THE INVENTION 1. Field of the Invention

The present invention relates to a method for updating a factor graph ofan a posteriori probability estimator.

More in particular, the present invention relates to a method forupdating a factor graph of an a posteriori probability estimator,wherein the repetition nodes, also called variable nodes or equalitynodes, and the sum nodes, or parity check nodes, of the factor graph areupdated with a message propagation algorithm.

The present invention further relates to an estimator, a decoder and areceiver implementing the method for updating a factor graph accordingto the invention.

2. Present State of the Art

Algorithms of the “Belief Propagation” type (“BP algorithms”), ormessage propagation algorithms, are known in the art.

In its most general form, the BP algorithm allows reconstructing themarginal probabilities for a set of random variables which describe asystem and which are connected by local constraints, expressed as jointprobability distributions P, by starting from a set of observations, inthe form of messages or verisimilitudes L, on a subset thereof.

FIG. 1 illustrates a node of a factor graph 1 that represents aprobability model, wherein the nodes 5 describe the local constraints,and the connections 3 between each node 5 describe the correspondingvariables.

In the BP algorithm, each connection 3 to a node 5 is associated with anincoming message 7 L(;I) and an outgoing message 9 L(;O), and each node5 of the factor graph 1 is replaced with a corresponding message updatenode.

The update rule for the node 5 is obtained by assuming statisticindependence of the incoming messages 7, through a marginalizationcarried out in accordance with the formula (1.1):

$\begin{matrix}{{L\left( {x_{j};O} \right)} \propto {\sum\limits_{\forall x_{\sim j}}^{\;}\;{{P(x)}{\sum\limits_{i \neq j}^{n}\;{{L\left( {x_{i};I} \right)}{\forall j}}}}}} & (1.1)\end{matrix}$

where x=(x₁, . . . , x_(n)) denotes a set of incoming variables at anode 5 and the symbol x_(˜j) denotes a subset of x obtained by excludingthe variable x_(j) from x.

The BP algorithm provides the exact a posteriori probabilities for allsystem variables in a finite number of steps when the graph is a tree(i.e. it contains no cycles), otherwise it gives a good approximationthereof.

As can be observed in the formula (1.1), the general complexity of thenode update is proportional to the product of the cardinalities of thealphabets of the incoming symbols at the node 5.

Furthermore, the messages 7,9 are represented by a function of theincoming symbols, and therefore they require, in order to be stored, amemory which is proportional to the cardinality of the symbol alphabets.

A particularly important example of application of the BP algorithm isobtained when the underlying model is a linear system.

In a linear system, the random variables are obtained as a linearcombination of other random variables.

For linear systems, the factor graph 1 is made up of three types ofnodes 5:

-   -   sum nodes: the sum of the incoming variables at a node 5 is        equal to zero;    -   repetition nodes (also called variable nodes or equality nodes):        two or more variables are equal;    -   scaling: the output is a scaled version of the input.

In this case, the rules for updating the messages at the nodescorrespond, respectively, to convolution, multiplication and scaling ofthe axes of the messages 7,9. In order to emphasize the properties ofthese update rules, the following notations will be adopted below forthe operations between messages 7,9:L ₁ +L ₂

L ₁(x)*L ₂(x)αL

L ₁(αx)L ₁ ·L ₂

L ₁(x)·L ₂(x)

where L₁ and L₂ are a first and a second messages relating to two inputvariables at a node 5, the symbol “*” indicates the convolutionoperator, and the symbol “

” indicates a definition.

Since convolution and product operations are associative, the updateoperation for a node 5 that involves more than three variables can becarried out by repeatedly applying the corresponding operations in theformula (1.2):

${{sum}\mspace{14mu}{node}\text{:}\mspace{14mu}{L\left( {x_{j};O} \right)}} = {- {\sum\limits_{i \neq j}^{n}\;{L\left( {x_{i};I} \right)}}}$${{repetition}\mspace{14mu}{node}\text{:}\mspace{14mu}{L\left( {x_{j};O} \right)}} = {\prod\limits_{i \neq j}^{n}\;{L\left( {x_{i};I} \right)}}$

It can be observed that the sign inversion simply corresponds to theoverturning of the axis of the messages: −L₁(x)

L₁(−x).

The complexity of the BP algorithm in this case increases linearly withthe number of variables involved in the node 5. In general, it can beshown that, in order to update all the messages for a sum node orrepetition node involving n variables, it is sufficient to carry out3·(n-2) corresponding binary update operations (convolution ormultiplication among messages).

The incoming and outgoing messages in the BP algorithm are normallyvectors whose size is equal to the cardinality of the alphabet of therespective variable.

In the following, the term message class will refer to a subset ofmessages which can be parameterized with a number of parameters smallerthan the dimensionality of the cardinality of the associated variable.In a BP algorithm operating on linear systems and continuous variables,it is not guaranteed that the output messages belong to the same classas that of the input messages, with the important exception of theGaussian message class.

The class of Gaussian (also referred to as Normal) messages, orG-messages, is defined as follows:

$\begin{matrix}{\mathcal{G}\left\{ {{{G\left( {\mu,K} \right)} \propto e^{{- \frac{K}{2}}{{x - \mu}}^{2}}},{K \in {\mathbb{R}}^{+}},{\mu \in {\mathbb{R}}}} \right\}} & (1.3)\end{matrix}$

where the two parameters K (called “concentration”, equal to the inverseof variance) and μ (mean) univocally define a member of the Gaussianmessage class.

G-messages form a closed set with respect to the convolution, productand scaling of the axes. In addition, the message update rules(convolution and product) translate into rules for updating theparameters of the class of G-messages as follows and as shown in FIG. 2:

$\begin{matrix}{{G_{1} + G_{2}} = {G_{3} = {G\;\left( {{\mu_{1} + \mu_{2}},\left( {K_{1}^{- 1} + K_{2}^{- 1}} \right)^{- 1}} \right)}}} & (1.4) \\{{G_{1} \cdot G_{2}} = {G_{3} = {G\left( {\frac{{\mu_{1}K_{1}} + {\mu_{2}K_{2}}}{K_{1} + K_{2}},{K_{1} + K_{2}}} \right)}}} & (1.5) \\{{\alpha\; G} = {{G\left( {{\alpha\;\mu},\frac{K}{\alpha^{2}}} \right)}.}} & (1.6)\end{matrix}$

The simplification found in the formulas (1.4), (1.5) and (1.6) allowsthe BP algorithm to be executed on factor graphs 1 with finitecomplexity because:

-   -   the messages 7,9 are only stored through the two parameters        (concentration and mean) that identify the member of the        Gaussian message class according to the formula (1.3):        otherwise, the message 7,9 would be a function of the continuous        variable x, and therefore would require infinite memory in order        to be stored;    -   the update operations are only executed on the parameters of the        Gaussian message class by using the formulas (1.4), (1.5) or        (1.6): in this case as well, the complexity of the convolution        and multiplication operations would be infinite for generic        messages 7,9.

For further details about the derivation of update rules for linearfactor graphs with Gaussian messages, see for example the article byLoeliger H.-A., Dauwels J., Junli Hu, Korl S., Li Ping, Kschischang F.R.: “The Factor Graph Approach to Model-Based Signal Processing”,Proceedings of the IEEE, vol. 95, no. 6, pp. 1295-1322, June 2007, whileEuropean patent application no. EP 2144413 illustrates an application ofthe algorithm described in the cited article for carrying out channelestimate in numerical receivers.

On the other hand, when the system variables are discrete and/orlimited, i.e. when discrete alphabets are used, the complexity andmemory requirement for message representation of the existing algorithmsgrow at least linearly with the cardinality of the message alphabets.

The BP algorithm thus becomes inefficient for high-cardinality finitealphabets.

A technical field where BP algorithms are used is channel coding withthe introduction of iterative algorithms for decoding very large randomcodes.

Very large random codes with binary alphabets allow reaching performancelevels close to the theoretical limit, but the complexity of theiroptimal decoding grows exponentially with the code size.

Instead, the complexity of the BP algorithm grows only linearly with thenumber of variables involved, and therefore ensures very goodperformance with reasonable complexity.

The use of discrete but not binary variables (cardinality M) at theencoder is desirable for several reasons.

Encoders with high spectral efficiency, which map sets of m=log₂ M bitscoded on high-cardinality constellations of the QAM or PAM type areoften used in telecommunication standards.

For this type of applications, non-binary encoders are the naturalchoice, and provide better performance than binary codes. However,non-binary encoders are not often employed in practice, mainly becausethe BP algorithm has a complexity which is at least linear with M andtherefore exponential with m.

For this reason, considerable efforts have been made in thetelecommunications industry to reduce the complexity of the BP algorithmfor non-binary codes, but the best algorithm known (based on FFT) has acomplexity of the order of O(M log M) and a memory requirementproportional to M.

For example, the article by Voicila A., Declercq D., Verdier F.,Fossorier M., Urard P., entitled “Low-complexity decoding for non-binaryLDPC codes in high order fields”, IEEE Transactions on Communications,vol. 58, no. 5, pp. 1365-1375, May 2010, describes an algorithm, calledEMS (Extended Min Sum), which further reduces this complexity, at theprice however of a degradation in the performance of the decoder, whileEuropean patent application no. EP2198523 represents an application ofthe method proposed in said article wherein, in order to execute the BPalgorithm, messages in the form of vectors are used and the update rulesare obtained by simplifying the convolution operations.

SUMMARY OF THE INVENTION

It is one object of the present invention to provide a method forupdating a factor graph of an a posteriori probability estimator whichrequires a certain quantity of memory for message representation whichis independent of the size of the alphabets in use.

It is another object of the present invention to provide a method forupdating a factor graph of an a posteriori probability estimator whichallows acting upon messages relating to limited and/or discretevariables.

It is a further object of the present invention to provide a method forupdating a factor graph of an a posteriori probability estimator whichis applicable for iterative decoding of a non-binary code with high orvery high cardinality.

These and other objects of the invention are achieved through a methodfor updating a factor graph of an a posteriori probability estimator asclaimed in the appended claims, which are an integral part of thepresent description.

The invention also relates to an estimator, a decoder and a receiverimplementing the method of the present invention.

In short, the subject of the present invention is a method for updatinga factor graph of an a posteriori probability estimator, whereinGaussian like messages are used at the sum and repetition nodes, andwherein discrete and/or limited variables are employed.

The messages are parameterized with their mean and concentration, andthe graph update operations have a complexity which does not depend onthe cardinality of the alphabets of the variables involved at the nodes.

Therefore, the messages are not represented as vectors whose length isequal to the cardinality of the associated variable, but through a fewparameters which identify them within the predefined class. Thisapproach makes both the computational complexity and the memoryrequirements of the algorithm independent of the cardinality of thevariables associated with the graph.

Furthermore, the method according to the present invention allows toprovide iterative decoders for non-binary linear channel encoders overrings, with a computational complexity and a memory requirementindependent of the dimension of the alphabets in use, thereby allowingfor the creation of decoders offering performance levels close to thetheoretical limit even for very high spectral efficiencies (tens orhundreds of bits per transmitted symbol), which cannot be obtained withany other prior art algorithm.

In one embodiment of the invention, the class of D-messages isintroduced, i.e. messages wherein the message variables are discrete andlimited. Said D-messages have low processing complexity and require asmall quantity of memory for storage, because the parameterizationthereof is independent of alphabet cardinality. D-messages thus open theway to intensive application of non-binary codes with high or even veryhigh cardinality in the telecommunications field.

Further features of the invention are set out in the appended claims,which are intended to be an integral part of the present description.

BRIEF DESCRIPTION OF THE DRAWINGS

The above objects will become more apparent from the following detaileddescription of a method for updating a factor graph of an a posterioriprobability estimator, with particular reference to the annexeddrawings, wherein:

FIG. 1 shows a node of a prior art factor graph representing aprobabilistic model;

FIG. 2 shows a factor graph wherein Gaussian messages are exchanged inaccordance with the prior art;

FIG. 3 shows a factor graph wherein wrapped messages are exchanged inaccordance with a first embodiment of the method of the presentinvention;

FIG. 4 shows a factor graph wherein sampled messages are exchanged inaccordance with a second embodiment of the method of the presentinvention;

FIGS. 5 and 6 show respective block diagrams illustrating the blocks andoperations required for updating the messages at the sum nodes inaccordance with a first and a second variants of said second embodiment;

FIG. 7 shows a factor graph wherein sampled and wrapped messages areexchanged in accordance with a third embodiment of the method of thepresent invention;

FIG. 8 shows a relation between Gaussian messages, wrapped messages,sampled messages, and sampled and wrapped messages;

FIG. 9 shows a table that summarizes the formulas that can be used inaccordance with the method of the present invention for wrappedmessages, sampled messages, and sampled and wrapped messages;

FIG. 10 shows a digital transmission and reception system thatimplements the method of the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

With reference to FIG. 3, there is shown a factor graph 10 of anestimator that comprises memory means 12 adapted to store incoming andoutgoing messages at sum nodes 11 and repetition nodes 13 of the factorgraph 10.

In a first embodiment of the method according to the invention, theoperation “modulo M”, where M∈

, is carried out on the variables a at the output of the sum nodes 11 ofthe factor graph 10, i.e. new variables a_(w)=a mod M are taken intoaccount.

The module operation carried out on the output variables induces awrapping of the corresponding messages L_(a).

The messages L_(a) _(w) (x) of the variables a_(w) are obtained fromthose relating to the variables a as follows:

$\begin{matrix}{{{L_{a_{w}}(x)} = {{\sum\limits_{i}^{\;}\;{L_{a}\left( {x - {iM}} \right)}} = {{s_{M}(x)}*{L_{a}(x)}{W\left\lbrack {L_{a}(x)} \right\rbrack}}}},} & (1.7)\end{matrix}$where the pulse train

${s_{M}(x)}{\sum\limits_{i = {- \infty}}^{\infty}\;{\delta\left( {x - {iM}} \right)}}$and a wrapping operator W[.] are introduced.

The wrapping of the messages is transparent with respect to the sumoperation, because the operator W switches with the convolution product:W[L ₁]+W[L ₂]=W[L ₁ +L ₂]

In particular, this introduces the class of wrapped Gaussian messages(WG-messages), also referred to as wrapped normal messages, which isdefined by applying the wrapping operator W to the G-messages of theformula (1.3):

=W[

]={W[G(μ,K,x)],K∈

⁺,μ∈[0,M]}

This class is still parameterized with the concentration and mean(module M) of the corresponding G-messages.

The class

of WG-messages is closed with respect to the sum operation:W[G ₁]+W[G ₂]=W[G ₁ +G ₂]=W[G ₃]∈

.  (1.8)

On the other hand, the wrapping has a non-transparent effect at therepetition nodes, since in general it does not switch withmultiplication:W[L ₁]·W[L ₂]≠W[L ₁ ·L ₂]  (1.9)

This effect is due to the fact that, when the original messages do nothave support strictly limited in [0,M], the replicas, or copies, inducedby the wrapping of the message are superimposed (aliasing effect):

$\begin{matrix}{{\sum\limits_{j}^{\;}\;{{L_{1}\left( {x - {jM}} \right)}{\sum\limits_{i}^{\;}\;{L_{2}\left( {x - {iM}} \right)}}}} = {{\sum\limits_{j}^{\;}\;{{L_{1}\left( {x - {jM}} \right)}{L_{2}\left( {x - {jM}} \right)}}} + \underset{\underset{aliasing}{︸}}{\sum\limits_{i \neq j}^{\;}\;{{L_{1}\left( {x - {jM}} \right)}{L_{2}\left( {x - {iM}} \right)}}}}} & (1.10)\end{matrix}$

In particular, the class of WG-messages is no longer closed for thisoperation, because G-messages do not have limited support in theinterval [0,M]:W[G ₁]·W[G ₂]∉

  (1.11)

In order to execute a BP algorithm to update the parameters of the classof WG-messages, it is therefore appropriate to find an approximation ofthe product of (1.11) in the class

of WG-messages.

To this end, the class of Von Mises messages (V-messages) is taken intoaccount:

{V(μ,K _(v))∝e ^(K) ^(v) ^(cos(A(x−μ))) ,K _(v)∈

⁺[0,M]}  (1.12)

where K_(v) is the concentration and μ is the mean of the V-message, andwhere the quantity

$A\frac{2\;\pi}{M}$is defined.

The class of V-messages is closed with respect to the product∀V ₁ ,V ₂ ∈

,V ₁ ·V ₂ =V ₃∈

  (1.13)

The parameters of the output V-message V₃ can be obtained as follows:

$\begin{matrix}{{\mu_{3} = {\frac{1}{A}{\arctan\left( \frac{{K_{v,1}{\sin\left( {A\;\mu_{1}} \right)}} + {K_{v,2}{\sin\left( {A\;\mu_{2}} \right)}}}{{K_{v,1}{\cos\left( {A\;\mu_{1}} \right)}} + {K_{v,2}{\cos\left( {A\;\mu_{2}} \right)}}} \right)}}}{K_{v,3} = {\sqrt{K_{v,1}^{2} + K_{v,2}^{2} + {2\; K_{v,1}K_{v,2}{\cos\left( {A\left( {\mu_{1} - \mu_{2}} \right)} \right)}}}.}}} & (1.14)\end{matrix}$

It is also possible to approximate a member of the class of WG-messageswith mean μ with a member of the class of V-messages with the same meanby applying the following transformations between the respectiveconcentrations, as known, for example, from the book written by Evans M,Hastings N. and Peacock B.: “Von Mises Distribution”, chapter 41,Statistical Distributions, third edition, New York, Wiley, 2000:

$\begin{matrix}{{{W\;\left\lbrack {G\left( {\mu,K_{w}} \right)} \right\rbrack} \approx {V\;\left( {\mu,K_{v}} \right)}}{K_{w} = {{A^{2}\left( {2\;{\log\left( \frac{I_{0}\left( K_{v} \right)}{I_{1}\left( K_{v} \right)} \right)}} \right)}^{- 1}A^{2}{f\left( K_{v} \right)}}}} & (1.15) \\{{{V\;\left( {\mu,K_{v}} \right)} \approx {W\;\left\lbrack {G\left( {\mu,K_{w}} \right)} \right\rbrack}}{K_{v} = {f^{- 1}\left( \frac{K_{w}}{A^{2}} \right)}}} & (1.16)\end{matrix}$

where the function ƒ(•) has been defined in order to make the formulasimpler, and where K_(v) and K_(w) are, respectively, the concentrationsof a V-message and of a WG-message approximating it.

In the formula (1.15), moreover, I_(i)(x) indicates the modified Besselfunctions of order i.

In order to find a member of the class

of WG-messages which approximates the formula (1.11), it is thereforenecessary to:

-   -   approximate the two WG-messages of the formula (1.11) with two        V-messages by using the formula (1.16);    -   calculate the product between the two V-messages with the        formula (1.14) and approximate again the resulting V-message        with a WG-message by using the formula (1.15).

In brief

$\begin{matrix}{{{{W\;\left\lbrack G_{1} \right\rbrack}{W\;\left\lbrack G_{2} \right\rbrack}} \approx {W\;\left\lbrack G_{3} \right\rbrack}}{where}{K_{v,i}{f^{- 1}\left( {K_{w,i}/A^{2}} \right)}}\;{{i = 1},2}} & (1.17) \\{{\mu_{3} = {\frac{1}{A}{\arctan\left( \frac{{K_{v,1}{\sin\left( {A\;\mu_{1}} \right)}} + {K_{v,2}{\sin\left( {A\;\mu_{2}} \right)}}}{{K_{v,1}{\cos\left( {A\;\mu_{1}} \right)}} + {K_{v,2}{\cos\left( {A\;\mu_{2}} \right)}}} \right)}}}{K_{w,3} = {A^{2}{{f\left( \sqrt{K_{v,1}^{2} + K_{v,2}^{2} + {2\; K_{v,1}K_{v,2}{\cos\left( {A\left( {\mu_{1} - \mu_{2}} \right)} \right)}}} \right)}.}}}} & (1.18)\end{matrix}$

The approximation of the formula (1.18) is preferably made when thealiasing effect induced by the wrapping of the formula (1.10) is notnegligible, i.e. when the concentration of either message is notsufficiently small, i.e. it is above a threshold that needs to beoptimized depending on the specific application. Otherwise, one maydirectly use the repetition node update formula (1.5) for non-wrappedvariables:

$\begin{matrix}{{{W\;\left\lbrack G_{1} \right\rbrack} \cdot {W\;\left\lbrack G_{2} \right\rbrack}} \approx {W\left\lbrack {G\left( {{\frac{{{\overset{\_}{\mu}}_{1}K_{1}} + {{\overset{\_}{\mu}}_{2}K_{2}}}{K_{1} + K_{2}}❘\mspace{14mu}{{mod}\mspace{14mu} M}},K_{1},K_{2}} \right)} \right\rbrack}} & (1.19)\end{matrix}$

where the two means μ ₁, μ ₂ in the formula (1.19) must be chosen, inthis case, among all the replicas (or copies) introduced by thewrapping, so that the distance is minimal. In other terms, μ ₁=μ₁+k₁Mand μ ₁=μ₂+k₂M and the integers k₁, k₂ are such that |μ ₁−μ ₂| isminimum.

The following will describe a second embodiment of the method accordingto the invention, wherein the system variables are assumed to bediscrete, i.e. x∈

⊂

.

With reference to FIG. 4, there is shown a factor graph 10′ of anestimator that comprises memory means 12′ adapted to store incoming andoutgoing messages at sum nodes 11′ and repetition nodes 13′ of thefactor graph 10′.

This a priori knowledge can be imposed in the BP algorithm by samplingthe corresponding messages, which operation corresponds to multiplyingan incoming message at a node 11′,13′ by a pulse train s₁(x):S[L(x)]

s ₁(x)·L(x)  (1.20)

where the sampling operator S[•] has been introduced.

The behaviour of the sampling operator S[•] with respect to theoperations at the nodes 11′,13′ of the graph 10′ of a linear system iscomplementary to that of the wrapping operator W[•] of the firstembodiment of the method according to the present invention.

The sampling of the messages is, in fact, transparent to the update ofthe repetition nodes 13′, but it is not with respect to the sum nodes11′:S[L ₁]·S[L ₂]=S[L ₁ ·L ₂]S[L ₁]+S[L ₂]≠S[L ₁ +L ₂]  (1.21)

The class of sampled Gaussian messages (SG-messages) is thus introduced,which is obtained by applying the sampling operator S[•] to G-messages.

The class

S[g] of sampled Gaussian messages is closed with respect to the product,but it is not with respect to the convolution:S[G ₁]·S[G ₂]=S[G ₁ ·G ₂]=S[G ₃]  (1.22)S[G _(1]+) S[G _(2]≠) S[G ₃]  (1.23)

In this case, in order to be able to proceed in the BP algorithm by onlyusing the parameters of the SG-messages, it is necessary to find anapproximation of the formula (1.23) in the class of SG-messages.

In order to proceed, one must take into consideration that:

-   -   the convolution product of SG-messages corresponds to the        product of their characteristic functions;        {S[G ₁]+S[G ₂]}=        {S[G ₁]]}·        {S[G ₂]]}  (1.24)    -   the characteristic functions of SG-messages are WG-messages with        a purely imaginary mean value and a unitary period multiplied by        a complex exponential;

${\mathcal{F}\left\{ {S\left\lbrack {G\left( {\mu,K,x} \right)} \right\rbrack} \right\}} \propto {\sum\limits_{n = {- \propto}}^{\infty}\; e^{{{- \frac{2\;\pi^{2}}{K}}{({p - n})}^{2}} - {j\; 2\;{\pi{({l + \alpha})}}{({p - n})}}}} \propto {e^{{- j}\; 2\;\pi\;{lp}}{W\left\lbrack {G\left( {{{- j}\frac{K\;\alpha}{2\;\pi}},\frac{4\;\pi^{2}}{K},p} \right)} \right\rbrack}}$

where the following has been defined:l

[μ]∈

integer closest to μα

μ−l∈[−0.5,0.5]

-   -   the approximations of the formulas (1.15) and (1.16) between        members of the class of V-messages and those of the class of        WG-messages are also valid when the WG-messages have an        imaginary mean value;

${W\left\lbrack {G\left( {p,{{- j}\frac{K_{w,i}\alpha_{i}}{2\;\pi}},\frac{4\;\pi^{2}}{K_{w,i}}} \right)} \right\rbrack} \approx {{V\left( {p,{{- j}\frac{K_{w,i}\alpha_{i}}{2\;\pi}},{f^{- 1}\left( \frac{1}{K_{w,i}} \right)}} \right)}.}$

-   -   the product of V-messages with imaginary mean value is still        closed, and the transformations that must be carried out in        order to obtain the parameters of the output V-message are        obtained from the formula (1.14) by transforming the        trigonometric functions with purely imaginary argument into        suitable hyperbolic functions,

$\begin{matrix}{{{V\;\left( {x,{{- j}\frac{\mu_{1}}{2\;\pi}},K_{v,1}} \right)V\;\left( {x,{{- j}\frac{\mu_{2}}{2\;\pi}},K_{v,2}} \right)} = {V\;\left( {x,{{- j}\frac{\mu_{3}}{2\;\pi}},K_{v,3}} \right)}}{\mu_{3} = {{\frac{1}{2}\log\;\frac{{K_{v,1}e^{\mu_{1}}} + {K_{v,2}e^{\mu_{2}}}}{{K_{v,1}e^{- \mu_{1}}} + {K_{v,2}e^{- \mu_{2}}}}K_{v,3}} = \sqrt{K_{v,1}^{2} + K_{v,1}^{2} + {2\; K_{v,2}K_{v,2}\cosh\;\left( {\mu_{1} - \mu_{2}} \right)}}}}} & (1.25)\end{matrix}$

where the formula (1.25) has been obtained by using the identity

${{arc}\;{\tanh(x)}} = {\frac{1}{2}{\log\left( \frac{1 + x}{1 - x} \right)}}$

On these grounds, the approximation of the formula (1.23) can be donethrough the following steps:

-   -   approximating the characteristic functions of the two        SG-messages (WG-messages with imaginary mean value and        concentrations proportional to the inverse of the original        concentrations) with V-messages;    -   calculating the product between said V-messages with imaginary        mean value by using the formula (1.25);    -   approximating the result with a WG-message that represents the        characteristic function of the SG-message which approximates the        formula (1.23).

In brief

$\begin{matrix}{{{{S\left\lbrack G_{1} \right\rbrack} + {S\left\lbrack G_{2} \right\rbrack}} \approx {S\left\lbrack G_{3} \right\rbrack}}{where}{K_{v,i} = {f^{- 1}\left( {1/K_{w,i}} \right)}}{\mu_{i} = \left( {l_{i} + \alpha_{i}} \right)}{\gamma_{i} = {\alpha_{i}K_{w,i}}}{\gamma_{3} = {\frac{1}{2}\log\frac{{K_{v,1}e^{\gamma_{1}}} + {K_{v,2}e^{\gamma_{2}}}}{{K_{v,1}e^{- \gamma}} + {K_{v,2}e^{- \gamma_{2}}}}}}{l_{3} = {l_{1} + l_{2}}}{K_{w,3} = {1/{f\left( \sqrt{K_{v,1}^{2} + K_{v,1}^{2} + {2\; K_{v,2}K_{v,2}{\cosh\left( {\gamma_{1} - \gamma_{2}} \right)}}} \right)}}}} & (1.26)\end{matrix}$

FIG. 5 shows a block diagram which illustrates the operations that mustbe carried out in order to obtain the formula (1.26).

The formula (1.26) should preferably be used when the concentration ofthe characteristic functions of the messages is sufficiently high, i.e.above a threshold that must be optimized for each specific application,and therefore for small message concentration values. Otherwise, theupdate formula becomes the formula (1.4):

$\begin{matrix}{{{S\;\left\lbrack G_{1} \right\rbrack} + {S\;\left\lbrack G_{2} \right\rbrack}} \approx {S\;\left\lbrack {G\left( {{\mu_{1} + \mu_{2}},\left( {K_{w,1}^{- 1} + K_{w,2}^{- 1}} \right)^{- 1}} \right)} \right\rbrack}} & (1.27)\end{matrix}$

When concentration values are small, the correspondence between theconcentrations of V-messages and the concentrations of WG-messages ofthe formula (1.16) takes a logarithmic trend. The followingapproximations apply:

$\begin{matrix}{{{{f\;(x)} \approx {- \frac{1}{2\;\log\;\left( {x/2} \right)}}},{x < 1}}{{{f^{- 1}(x)} \approx {2\; e^{- \frac{1}{2\; x}}}},{x < 0.72}}} & (1.28)\end{matrix}$

It is therefore advantageous to introduce the concentration logarithmκ_(v,i)

log K_(v,i)−log (2) for V-messages and to derive the update equations(1.26) by using this quantity.

By introducing the associative operator max*

log Σ exp to simplify the notations, the update equations can bere-written as follows:

$\begin{matrix}{\mspace{79mu}{{\kappa_{v,i} = {{- K_{w,i}}/2}}\mspace{79mu}{\mu_{i} = \left( {l_{i} + \alpha_{i}} \right)}\mspace{79mu}{\gamma_{i} = {\alpha_{i}K_{w,i}}}{K_{w,3} = {- {\max^{*}\left( {{{2\;\kappa_{v,1}} + {2\;\kappa_{v,1}}},{\kappa_{v,1} + \kappa_{v,2} + {\max^{*}\left( {{\gamma_{1} - \gamma_{2}},{\gamma_{2} - \gamma_{1}}} \right)}}} \right)}}}{\gamma_{3} = {\frac{1}{2}\left( {{\max^{*}\left( {{\kappa_{v,1} + \gamma_{1}},{\kappa_{v,2} + \gamma_{2}}} \right)} - {\max^{*}\left( {{\kappa_{v,1} - \gamma_{1}},{\kappa_{v,2} - \gamma_{2}}} \right)}} \right)}}}} & (1.29)\end{matrix}$

This update operation, which is equivalent to the formula (1.26), onlyrequires sums and max* operators, and is therefore particularly usefulwhen implementing the method of the present invention.

FIG. 6 shows a block diagram which illustrates the operations requiredfor obtaining the formula (1.29).

The following will describe a third embodiment of the method accordingto the invention, wherein variables are made discrete and are wrappedwith integer period M.

With reference to FIG. 7, there is shown a factor graph 10″ of anestimator that comprises memory means 12″ adapted to store incoming andoutgoing messages at sum nodes 11″ and repetition nodes 13″ of thefactor graph 10″.

In this third embodiment, the sampling operators S[•] and the wrappingoperators W[•] switch, and it is possible to introduce the class ofdigital messages (D-messages)

S[W[

]]=W[S[

]]

which can still be parameterized through the mean and the variance ofthe corresponding Gaussian message.

The relation among WG-messages, SG-messages and D-messages is shown inFIG. 8.

The class of D-messages is not closed either with respect to the sumnodes 11″ or with respect to the repetition nodes 13″. However, it isstill possible to apply the approximations introduced with reference toWG-messages and SG-messages:W[S[G ₁]]+W[S[G ₂]]=W[S[G ₁]+S[G ₂]]≈W[S[G ₃]]  (1.30)S[W[G ₁]]·S[W[G ₂]]=S[W[G ₁]·W[G ₂]]≈S[W[G ₃]]  (1.31)

Discrete variables wrapped with integer period M are variables withfinite cardinality M which are obtained, for example, in linear encodersover rings, where inputs are discrete variables in [0,M−1] and sums aremade with module M. Said class of linear encoders is a generalization ofbinary codes (M=2).

The method of said third embodiment, which uses D-messages, thus allowsfor iterative decoding of linear codes over rings. It should be pointedout, however, that the variable scaling operation (product by a scalar)is not possible.

The complexity of the corresponding iterative decoder is thereforeindependent of M.

An overview of the formulas that can be used in accordance with themethod of the present invention for WG, SG and D type messages is shownin FIG. 9.

The decoder implementing the method according to one of the threeabove-described embodiments of the invention comprises an estimator.

FIG. 10 illustrates an example of a digital transmission and receptionsystem which advantageously uses the a posteriori probability estimatorof the present invention. In particular, FIG. 10 shows a transmissionand reception system that employs linear codes over rings with M-PAMmodulations and a receiver comprising an a posteriori probabilityestimator as shown in the third embodiment of the invention.

An encoder 31 with rate K/N over a ring with cardinality M generatesblocks of N integer symbols c_(k) in the interval [0,M−1] starting fromblocks of K symbols u_(l) of the same cardinality. The symbols c_(k) aretransmitted through an M-PAM modulation on an additive Gaussian channelby means of the correspondence x(c) that defines the M-PAM typemodulator 32.

The task of the receiver is to estimate the symbols transmitted atmaximum a posteriori probability (MAP). To do so, it is necessary tocalculate the a posteriori probability of the latter and decide for thesymbol that maximizes it.

Since the model that describes an encoder over rings is a linear modelwhich only involves repetitions and sums modulo M and the input messagesto the estimator 34 are sampled and Gaussian messages, it is possible touse the version that processes D-messages of the third embodiment of thepresent invention through a processing module 33.

The receiver of FIG. 10, based on the a posteriori probability estimatorof the present invention, has a complexity and memory requirements whichare independent of M, and hence independent of the spectral efficiency ηof the system, unlike prior art receivers, wherein the complexity of theestimator is at least linear with M and therefore exponential with thespectral efficiency of the system.

The method and the estimator according to the present invention may alsobe used with the method according to the first and second embodiments ofthe present invention.

The features and advantages of the present invention are apparent fromthe above description.

The method according to the present invention advantageously introducesa new family of methods of propagation of messages for discrete and/orwrapped variables.

The method for updating a factor graph of an a posteriori probabilityestimator according to the present invention advantageously requires aquantity of memory for message representation which is constant andindependent of the cardinality of the variables' alphabets in use.

Another advantage is that the method of the present invention allowsacting upon messages wherein variables are limited and/or discrete.

A further advantage is that the method of the present invention allowscreating iterative decoders for non-binary codes with high or very highcardinality.

Finally, the method of the present invention can advantageously be usedfor at least the following applications:

-   -   generation of iterative decoding algorithms for        telecommunications channel codes (LDPC, turbo codes and the        like);    -   MIMO receivers, source coding and receivers for digital        telecommunications;    -   artificial intelligence and neural networks;    -   optimization and modelling;    -   any application using BP algorithms on linear systems also        involving discrete and/or finite support variables and sums        modulo M.

The method for updating a factor graph of an a posteriori probabilityestimator described herein by way of example may be subject to manypossible variations without departing from the novelty spirit of theinventive idea; it is also clear that in the practical implementation ofthe invention the illustrated details may have different shapes or bereplaced with other technically equivalent elements.

It can therefore be easily understood that the present invention is notlimited to a method for updating a factor graph of an a posterioriprobability estimator, but may be subject to many modifications,improvements or replacements of equivalent parts and elements withoutdeparting from the inventive idea, as clearly specified in the followingclaims.

The invention claimed is:
 1. A computer-implemented method for updatinga factor graph of an a posteriori probability estimator which reducescomputing complexity and reduces memory requirements, said factor graphcomprising at least one repetition node and at least one sum node,wherein at least two connections are associated with each node, andwherein each connection is associated with an incoming message at saidnode and with an outgoing message from said node; said methodcomprising: a) storing the nodes' incoming and outgoing messages intomemory of said estimator as messages belonging to one same class ofwrapped Gaussian messages, wherein the messages are represented with twoparameters including a mean μ and a concentration K of the correspondingwrapped Gaussian distribution; b) updating said node of said factorgraph by using a resulting message belonging to said class of incomingmessages, said resulting message being obtained by processing saidincoming wrapped Gaussian messages; wherein said wrapped Gaussiandistribution describes a random variable which takes values in a finiterange and is parametrized as follows:${{W\left( {\mu,K,x} \right)} = {\sum\limits_{i = {- \infty}}^{\infty}e^{{- \frac{K}{2}}{({x - {iM} - \mu})}^{2}}}},{x \in \left\lbrack {0,M} \right\rbrack}$and wherein a wrapping period is M, wherein when said node is arepetition node, corresponding to the multiplication of thedistributions, the processing of the resulting message varies dependingon the concentration parameter of said incoming wrapped Gaussianmessages as follows: when the concentration is larger than a thresholdsaid resulting message has a mean of${\overset{\_}{\mu}}_{3} = {\frac{{{\overset{\_}{\mu}}_{1}K_{1}} + {{\overset{\_}{\mu}}_{2}K_{2}}}{K_{1} + K_{2}}❘}$mod M and a concentration of K₃=K₁+K₂, where μ ₁ and K₁ are,respectively, the means and the concentration of a first incomingwrapped Gaussian message at said repetition node, μ ₂ and K₂ are,respectively, the mean and the concentration of a second incomingwrapped Gaussian message at said repetition node, and M is the wrappingperiod of said incoming wrapped Gaussian messages, and wherein saidmeans μ ₁ and μ ₂ are chosen among all possible values of μ ₁=μ₁+k₁M andμ ₁=μ₂+k₂M so that |μ ₁−μ ₂| is minimum, and when the concentration issmaller than said threshold, the resulting message has a mean μ₃ of$\frac{1}{A}{\arctan\left( \frac{{K_{v,1}{\sin\left( {A\;\mu_{1}} \right)}} + {K_{v,2}{\sin\left( {A\;\mu_{2}} \right)}}}{{K_{v,1}{\cos\left( {A\;\mu_{1}} \right)}} + {K_{v,2}{\cos\left( {A\;\mu_{2}} \right)}}} \right)}$and a concentration K_(w,3) of A²f(√{square root over (K_(v,1) ²+K_(v,2)²+2K_(v,1)K_(v,2) cos (A(μ₁−μ₂)))}), where K_(v, i)f⁻¹(K_(w, i)/A²)i = 1, 2 ${A\frac{2\;\pi}{M}},$ wherein when said node is a sum node,corresponding to a convolution of the distributions, said resultingmessage has a mean which is the sum of the means of a first and a secondincoming wrapped Gaussian messages μ₃=μ₁+μ₂, and the concentration is(K₁ ⁻¹+K₂ ⁻¹), where K₁ is a concentration of said first incomingwrapped Gaussian message and K₂ is a concentration of said secondincoming wrapped Gaussian message.
 2. The method according to claim 1,wherein said message class of said nodes' incoming and outgoing messagescomprises the sampled and wrapped Gaussian messages, wherein thewrapping period M is an integer, and wherein said sampled and wrappedGaussian distribution describes a random variable which takes onlyinteger values in a finite range and is parametrized as follows${{{WG}\left( {\mu,K} \right)} = {\sum\limits_{j = {- \infty}}^{\infty}e^{{- \frac{K}{2}}{({i - {jM} - \mu})}^{2}}}},{i \in {\left\lbrack {0,1,{{\ldots\mspace{14mu} M} - 1}} \right\rbrack.}}$3. A computer-implemented method for updating a factor graph of an aposteriori probability estimator which reduces computing complexity andreduces memory requirements, said factor graph comprising at least onerepetition node and at least one sum node, wherein at least twoconnections are associated with each node, and wherein each connectionis associated with an incoming message at said node and with an outgoingmessage from said node; said method comprising: a) storing the nodes'incoming and outgoing messages into memory of said estimator as messagesbelonging to one same class of sampled Gaussian messages, representedwith a mean and a concentration of a corresponding sampled Gaussiandistribution; b) updating said nodes of said factor graph by using aresulting message belonging to said same class of incoming sampledGaussian messages, said resulting message being obtained by processingsaid incoming sampled Gaussian messages; wherein said sampled Gaussiandistribution describes a random variable which takes only integer valuesand is parametrized as follows${{G\left( {\mu,K} \right)} = {{e^{{- \frac{K}{2}}{({i - \mu})}^{2}}i} \in Z}},$wherein: when said node is a repetition node said resulting message hasa mean of $\frac{{\mu_{1}K_{1}} + {\mu_{2}K_{2}}}{K_{1} + K_{2}}$ and aconcentration of K₁+K₂, where μ₁ and K₁ are, respectively, a mean and aconcentration of a first incoming sampled Gaussian message, and μ₂ andK₂ are, respectively, a mean and a concentration of a second incomingsampled Gaussian message, when said node is a sum node the updating ofthe resulting message varies depending on the concentration parameter ofsaid incoming sampled Gaussian messages, wherein: when the concentrationis smaller than a threshold said resulting message has a mean which isthe sum of the means of a first and a second incoming sampled Gaussianmessages, and the concentration is (K₁ ⁻¹+K₂ ⁻¹)⁻¹, wherein K₁ is theconcentration of said first incoming sampled Gaussian message and K₂ isthe concentration of said second incoming sampled Gaussian message, andwhen the concentration is larger than said threshold said resultingmessage has a mean of μ₃ and a concentration of K_(w,3), obtainable fromthe following formulas: K_(v, i) = f⁻¹(1/K_(w, i))μ_(i) = (l_(i) + α_(i)) γ_(i) = α_(i)K_(w, i)$\gamma_{3} = {\frac{1}{2}\log\frac{{K_{v,1}e^{\gamma_{1}}} - {K_{v,2}e^{\gamma_{2}}}}{{K_{v,1}e^{- \gamma_{1}}} - {K_{v,2}e^{- \gamma_{2}}}}}$l₃ = l₁ + l₂$K_{w,3} = {1/{f\left( \sqrt{K_{v,1}^{2} + K_{v,1}^{2} + {2\; K_{v,2}K_{v,2}{\cosh\left( {\gamma_{1} - \gamma_{2}} \right)}}} \right)}}$where μ₁ and K₁ are, respectively, a mean and a concentration of a firstincoming sampled Gaussian message at said sum node, and μ₂ and K₂ are,respectively, a mean and a concentration of a second incoming sampledGaussian message at said sum node.
 4. The method according to claim 3,wherein said node is a sum node and said concentration is larger than athreshold, wherein said resulting message has a mean of μ₃ and aconcentration of K_(w,3), obtainable from the following formulas:     κ_(v, i) = −K_(w, i)/2      μ_(i) = (l_(i) + α_(i))     γ_(i) = α_(i)K_(w, i)K_(w, 3) = −max^(*)(2 κ_(v, 1), 2 κ_(v, 1), κ_(v, 1) + κ_(v, 2) + max^(*)(γ₁ − γ₂, γ₂ − γ₁))$\gamma_{3} = {\frac{1}{2}\left( {{\max^{*}\left( {{\kappa_{v,1} + \gamma_{1}},{\kappa_{v,2} + \gamma_{2}}} \right)} - {\max^{*}\left( {{\kappa_{v,1} - \gamma_{1}},{\kappa_{v,2} - \gamma_{2}}} \right)}} \right)}$where μ₁ and K₁ are, respectively, the mean and the concentration of afirst incoming sampled Gaussian message at said sum node, and μ₂ and K₂are, respectively, the mean and the concentration of a second incomingsampled Gaussian message at said sum node, and wherein the operatormax*≙ log Σ exp has been introduced.
 5. The method according to claim 3,wherein said message class of said nodes' incoming and outgoing messagescomprises the sampled and wrapped Gaussian messages, wherein thewrapping period M is an integer, and wherein said sampled and wrappedGaussian distribution describes a random variable which takes onlyinteger values in a finite range and is parametrized as follows${{{WG}\left( {\mu,K} \right)} = {\sum\limits_{j = {- \infty}}^{\infty}e^{{- \frac{K}{2}}{({i - {jM} - \mu})}^{2}}}},{i \in {\left\lbrack {0,1,{{\ldots\mspace{14mu} M} - 1}} \right\rbrack.}}$6. The method according to claim 4, wherein said message class of saidnodes' incoming and outgoing messages comprises the sampled and wrappedGaussian messages, wherein the wrapping period M is an integer, andwherein said sampled and wrapped Gaussian distribution describes arandom variable which takes only integer values in a finite range and isparametrized as follows${{{WG}\left( {\mu,K} \right)} = {\sum\limits_{j = {- \infty}}^{\infty}e^{{- \frac{K}{2}}{({i - {jM} - \mu})}^{2}}}},{i \in {\left\lbrack {0,1,{{\ldots\mspace{14mu} M} - 1}} \right\rbrack.}}$7. A system for updating a factor graph of an a posteriori probabilityestimator which reduces computing complexity and reduces memoryrequirements, said factor graph comprising at least one repetition nodeand at least one sum node, wherein at least two connections areassociated with each node, and wherein each connection is associatedwith an incoming message at said node and with an outgoing message fromsaid node, the system comprising: one or more computer processors; andcomputer readable memory having stored therein computer executableinstructions which, when executed upon the one or more processors, causethe system to be configured to perform: a) storing the nodes' incomingand outgoing messages into memory of said estimator as messagesbelonging to one same class of wrapped Gaussian messages, wherein themessages are represented with two parameters including a mean μ and aconcentration K of the corresponding wrapped Gaussian distribution; b)updating said node of said factor graph by using a resulting messagebelonging to said class of incoming messages, said resulting messagebeing obtained by processing said incoming wrapped Gaussian messages;wherein said wrapped Gaussian distribution describes a random variablewhich takes values in a finite range and is parametrized as follows:${{W\left( {\mu,K,x} \right)} = {\sum\limits_{i = {- \infty}}^{\infty}e^{{- \frac{K}{2}}{({x - {iM} - \mu})}^{2}}}},{x \in \left\lbrack {0,M} \right\rbrack}$and wherein a wrapping period is M, wherein when said node is arepetition node, corresponding to the multiplication of thedistributions, the processing of the resulting message varies dependingon the concentration parameter of said incoming wrapped Gaussianmessages as follows: when the concentration is larger than a thresholdsaid resulting message has a mean of${\overset{\_}{\mu}}_{3} = {\frac{{{\overset{\_}{\mu}}_{1}K_{1}} + {{\overset{\_}{\mu}}_{2}K_{2}}}{K_{1} + K_{2}}❘}$mod M and a concentration of K₃=K₁+K₂, where μ ₁ and K₁ are,respectively, the means and the concentration of a first incomingwrapped Gaussian message at said repetition node, μ ₂ and K₂ are,respectively, the mean and the concentration of a second incomingwrapped Gaussian message at said repetition node, and M is the wrappingperiod of said incoming wrapped Gaussian messages, and wherein saidmeans μ ₁ and μ ₂ are chosen among all possible values of μ ₁=μ₁+k₁M andμ ₄=μ₂+k₂M so that |μ ₁−μ ₂| is minimum, and when the concentration issmaller than said threshold, the resulting message has a mean μ₃ of$\frac{1}{A}{\arctan\left( \frac{{K_{v,1}{\sin\left( {A\;\mu_{1}} \right)}} + {K_{v,2}{\sin\left( {A\;\mu_{2}} \right)}}}{{K_{v,1}{\cos\left( {A\;\mu_{1}} \right)}} + {K_{v,2}{\cos\left( {A\;\mu_{2}} \right)}}} \right)}$and a concentration K_(w,3) of A²f(√{square root over (K_(v,1) ²+K_(v,2)²+2K_(v,1)K_(v,2) cos (A(μ₁−μ₂)))}), where K_(v, i)f⁻¹(K_(w, i)/A²)i = 1, 2 ${A\frac{2\;\pi}{M}},$ wherein when said node is a sum node,corresponding to a convolution of the distributions, said resultingmessage has a mean which is the sum of the means of a first and a secondincoming wrapped Gaussian messages μ₃=μ₁+μ₂, and the concentration is(K₁ ⁻¹+K₂ ⁻¹)⁻¹, where K₁ is a concentration of said first incomingwrapped Gaussian message and K₂ is a concentration of said secondincoming wrapped Gaussian message.
 8. The system according to claim 7,wherein said message class of said nodes' incoming and outgoing messagescomprises the sampled and wrapped Gaussian messages, wherein thewrapping period M is an integer, and wherein said sampled and wrappedGaussian distribution describes a random variable which takes onlyinteger values in a finite range and is parametrized as follows${{{WG}\left( {\mu,K} \right)} = {\sum\limits_{j = {- \infty}}^{\infty}e^{{- \frac{K}{2}}{({i - {jM} - \mu})}^{2}}}},{i \in {\left\lbrack {0,1,{{\ldots\mspace{14mu} M} - 1}} \right\rbrack.}}$9. A system for updating a factor graph of an a posteriori probabilityestimator which reduces computing complexity and reduces memoryrequirements, said factor graph comprising at least one repetition nodeand at least one sum node, wherein at least two connections areassociated with each node, and wherein each connection is associatedwith an incoming message at said node and with an outgoing message fromsaid node; the system comprising: one or more computer processors; andcomputer readable memory having stored therein computer executableinstructions which, when executed upon the one or more processors,causes the system to be configured to perform: a) storing the nodes'incoming and outgoing messages into memory of said estimator as messagesbelonging to one same class of sampled Gaussian messages, representedwith a mean and a concentration of a corresponding sampled Gaussiandistribution; b) updating said nodes of said factor graph by using aresulting message belonging to said same class of incoming sampledGaussian messages, said resulting message being obtained by processingsaid incoming sampled Gaussian messages; wherein said sampled Gaussiandistribution describes a random variable which takes only integer valuesand is parametrized as follows${{G\left( {\mu,K} \right)} = {{e^{{- \frac{K}{2}}{({i - \mu})}^{2}}i} \in Z}},$wherein: when said node is a repetition node said resulting message hasa mean of $\frac{{\mu_{1}K_{1}} + {\mu_{2}K_{2}}}{K_{1} + K_{2}}$ and aconcentration of K₁+K₂, where μ₁ and K₁ are, respectively, a mean and aconcentration of a first incoming sampled Gaussian message, and μ₂ andK₂ are, respectively, a mean and a concentration of a second incomingsampled Gaussian message, when said node is a sum node the updating ofthe resulting message varies depending on the concentration parameter ofsaid incoming sampled Gaussian messages, wherein: when the concentrationis smaller than a threshold said resulting message has a mean which isthe sum of the means of a first and a second incoming sampled Gaussianmessages, and the concentration is (K₁ ⁻¹+K₂ ⁻¹)⁻¹, wherein K₁ is theconcentration of said first incoming sampled Gaussian message and K₂ isthe concentration of said second incoming sampled Gaussian message, andwhen the concentration is larger than said threshold said resultingmessage has a mean of μ₃ and a concentration of K_(w,3), obtainable fromthe following formulas: K_(v, i) = f⁻¹(1/K_(w, i))μ_(i) = (l_(i) + α_(i)) γ_(i) = α_(i)K_(w, i)$\gamma_{3} = {\frac{1}{2}\log\frac{{K_{v,1}e^{\gamma_{1}}} - {K_{v,2}e^{\gamma_{2}}}}{{K_{v,1}e^{- \gamma_{1}}} - {K_{v,2}e^{- \gamma_{2}}}}}$l₃ = l₁ + l₂$K_{w,3} = {1/{f\left( \sqrt{K_{v,1}^{2} + K_{v,1}^{2} + {2\; K_{v,2}K_{v,2}{\cosh\left( {\gamma_{1} - \gamma_{2}} \right)}}} \right)}}$where μ₁ and K₁ are, respectively, a mean and a concentration of a firstincoming sampled Gaussian message at said sum node, and μ₂ and K₂ are,respectively, a mean and a concentration of a second incoming sampledGaussian message at said sum node.
 10. The system according to claim 9,wherein said node is a sum node and said concentration is larger than athreshold, wherein said resulting message has a mean of μ₃ and aconcentration of K_(w,3), obtainable from the following formulas:     κ_(v, i) = −K_(w, i)/2      μ_(i) = (l_(i) + α_(i))     γ_(i) = α_(i)K_(w, i)K_(w, 3) = −max^(*)(2 κ_(v, 1), 2 κ_(v, 1), κ_(v, 1) + κ_(v, 2) + max^(*)(γ₁ − γ₂, γ₂ − γ₁))$\gamma_{3} = {\frac{1}{2}\left( {{\max^{*}\left( {{\kappa_{v,1} + \gamma_{1}},{\kappa_{v,2} + \gamma_{2}}} \right)} - {\max^{*}\left( {{\kappa_{v,1} - \gamma_{1}},{\kappa_{v,2} - \gamma_{2}}} \right)}} \right)}$where μ₁ and K₁ are, respectively, the mean and the concentration of afirst incoming sampled Gaussian message at said sum node, and μ₂ and K₂are, respectively, the mean and the concentration of a second incomingsampled Gaussian message at said sum node, and wherein the operatormax*≙ log Σ exp has been introduced.